Bayesian Neural Networks With Physics-Informed Priors With Application to Boundary Layer Velocity

Area of Science:

• Computational fluid dynamics
• Machine learning
• Bayesian inference

Background:

• Physics-informed neural networks (PINNs) integrate partial differential equations (PDEs) into neural network training.
• Current PINN approaches lack formal uncertainty quantification.
• Minimization problems in PINNs often lack robust uncertainty estimation.

Purpose of the Study:

• To develop a novel framework for uncertainty quantification in physics-informed neural networks.
• To integrate PDEs as prior information within a Bayesian neural network (BNN) framework.
• To enable physically consistent forecasts for complex systems with limited data.

Main Methods:

• Proposed a physics-informed prior (PIP)-BNN framework where PDEs act as priors.
• Calibrated the prior mean to resemble PDE solutions and used prior variance for confidence.
• Propagated PDE information to the posterior for uncertainty quantification.

Main Results:

• The PIP-BNN framework successfully quantified uncertainty in forecasts.
• Demonstrated effectiveness on simulated viscous fluid flow and experimental turbulent boundary layer data.
• Achieved physically consistent forecasts with significantly fewer observations compared to non-informed priors.

Conclusions:

• The PIP-BNN approach provides a robust method for uncertainty quantification in physics-informed machine learning.
• This framework enables accurate forecasting of complex systems by leveraging both data and prior physical knowledge.
• The method shows promise for applications in scientific modeling and engineering where data is scarce but physical laws are known.